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Full text of "Treatise On Analysis Vol-Ii"

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16 BAIRE'S THEOREM AND ITS CONSEQUENCES 83 (b) Deduce that, if A is a weakly closed convex set in E, and B is a weakly com- pact convex set disjoint from A, then there exists a weakly closed hyperplane H = {/e V :/(xo) = a} where XQ ^ 0, such that g(xQ) ;> oc for all g e A, and h(x0) ^ a for all h e B. (Consider the convex set A -f (—B).) (c) Deduce from (b) that for a vector subspace F not to be dense in V for the weak topology, it is necessary and sufficient that there should exist x0 ^ 0 in E such that = Oforall/eF. 14. In a Hilbert space E, let (xn) be a sequence which converges weakly to a point a. Show that, for each point b ^ a, lim mf\\xn- 61| > lim mf\\xm - a\\. (Expand ||*B - A||2 = \\(xn - a) + (« - 6)||2.) 16. BAIRE'S THEOREM AND ITS CONSEQUENCES (12.16.1) (Baire's theorem) Let E be a topological space in which every point has a neighborhood homeomorphic to a complete metric space. If(Un) is a sequence of dense open sets in E, then the intersection of the Un is dense in E. It is enough to prove that, for each x e E and each neighborhood V of x, the intersection of V and the Un is not empty. Hence we may assume that E itself is a complete metric space and (bearing in mind (3.14.5)) prove that the intersection G of the Un is nonempty. Let d be a distance defining the topology of E, with respect to which E is complete. We shall define by induction on n a sequence (xn) of points of E and a sequence (rn) of real numbers >0, as follows: x1 e U1; rn < l/n for each n ^ 1; the closed ball W(xn; rn) is contained in Un n B'(X,-i; /*„_!). This is possible because Un n B^^!; rn^a) is a nonempty open set in E, since \Jn is dense in E. Clearly we have d(xn, xn+p) g rn < l/n for each n g: 1 and each p > 0, and therefore (xn) is a Cauchy sequence in E. By hypothesis it converges to a point aeE, and since xn+pe B'(xn; rn) for each /?>0, we have a € B'(xn; rn) c Un for each n since R'(xn; rn) is closed in E. The point a there- fore belongs to G. Q.E.D. We shall apply Baire's theorem mainly when E is an open subspace of a complete metric space or a locally compact metrizable space (by virtue of (3.16.1)). In a topological space E, a subset A is said to be nowhere dense if the open set E - A is dense (or, equivalently, if A contains no nonempty open set, or if A has empty interior). For example, in a Hausdorff space E, a set {a} consistingoblem 10 to deduce that <D(~O and Q.(^f) are orthogonal supplementary